
A positively charged muon track is required to be reconstructed in the STRAW spectrometer with momentum in the range 5–70 GeV/\(c\) . The track's trajectory through the STRAW chambers and its extrapolation to the LKr calorimeter, CHOD and MUV3 should be within the geometrical acceptance of these detectors. The muon time is evaluated using the RICH and CHOD signals spatially associated with the track.

Particle identification criteria are applied to the STRAW track to suppress the backgrounds due to misidentification. The ratio of the energy deposited in the LKr calorimeter, \(E\) , to the momentum, \(p\) , measured by the STRAW spectrometer is required to be \(E/p<0.2\) . For tracks with momentum below 30 GeV/\(c\) , a particle identification algorithm is applied based on the RICH signal pattern within 3 ns of the CHOD time. In particular, tracks with momenta below the muon Cherenkov threshold must not be identified as positrons. At least one signal in the MUV3 detector must be within 3 ns of the muon time and spatially consistent with the projected track impact point in the MUV3 front plane.

Backgrounds from \(K^+\rightarrow \mu ^+\nu \)  decays upstream of the KTAG and \(\pi ^+\rightarrow \mu ^+\nu \)  decays upstream of GTK3, in coincidence with a beam pion or proton track in the GTK, are suppressed by requiring a kaon signal in the KTAG detector within 1 ns of the muon time.

The decay vertex is defined as the point of closest approach of the \(K^+\)  track in the GTK and the muon track in the STRAW, taking into account the stray magnetic field in the vacuum tank. Identification of the \(K^+\)  track in the GTK relies on the time difference, \(\Delta t_{\rm GK}\) , between a GTK track and the KTAG signal, and spatial compatibility of the GTK and STRAW tracks quantified by the distance, \(d\) , of closest approach. A discriminant \({\cal D}(\Delta t_{\rm GK}, d)\)  is defined using the \(\Delta t_{\rm GK}\)  and \(d\)  distributions measured with \(K^+\rightarrow \pi ^+\pi ^+\pi ^-\)  decays [1]}. Among GTK tracks with \(|\Delta t_{\rm GK}|<0.5\)  ns, the track of the parent kaon is assumed to be the one with the \({\cal D}\)  value most consistent with a \(K^+\rightarrow \mu ^+\)  decay. It is required that \(d<7\)  mm to reduce the background from upstream decays.

Background from \(K^+\rightarrow \mu ^+\nu \)  decays between KTAG and GTK3 with pileup in the GTK is suppressed by geometrical conditions. The reconstructed \(K^+\)  decay vertex is required to be located in the FV at a minimum distance from the start of the FV, varying from 8 m to 35 m depending on the angle between the \(K^+\)  momentum in the laboratory frame and the muon momentum in the \(K^+\)  rest frame.

Backgrounds from \(K^+\)  decays to multiple detectable particles are suppressed by veto conditions. The muon track must not form a vertex with any additional STRAW track segment. Energy deposits are not allowed in the LKr calorimeter that are spatially incompatible with the muon track within 12 ns of the muon time. No activity is allowed in the large-angle (LAV) or small-angle (SAC, IRC) photon veto detectors within 3 ns of the muon time, or in the CHANTI detector within 4 ns of the muon time. No more than two signals in the CHOD tiles within 6 ns of the muon time, and no more than three signals in the RICH PMTs within 3 ns of the muon time, spatially incompatible with the muon track, are allowed. Data loss due to the veto conditions from accidental activity (random veto) averaged over the data sample is measured to be about 30%.

